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In the St. Petersburg Paradox, which is more of a gambling game than a paradox, a balanced coin is tossed fairly until the first head appears. The gambler's winnings are based on the number of tosses that are made before the game ends. If a head appears on the first toss, the player wins $2. If not, the "kitty" is doubled to $4 -- the reward if a head appers on the second toss. The pot is doubled after every coin toss that results in a tail. The winnings are $2 raised to the power of the number of tosses until and including the first head. This procedure is more interesting when you think about what amount you would be willing to pay for the privilege of playing the game. The probability that n+1 tosses will occur before payment is the probability that there is a run of n tails and that the (n+1)st toss is a head or (1/2)n+1. The payoff for n+1 tosses is 2n+1. We calculate the player's expected receipts from the sum$2(1/2)+$22(1/2)2+$23(1/2)3+...=$1+1+1+...=$infinite.

Since the number of $1's in this sum is unlimited, the expected receipts from a play of this game are infinite! Whatever amount you were willing to play must have been a finite amount and therefore less than the expected receipts.

OK, this shows quite clearly, that "rational decision" in the case of gambling cannot be simply defined by this way of calculation. This should be quite obvious from an example like a 50-50 chance of winning $1,000,000 when investing $500,000, considering the possibylity of going broke and becoming unable to feed the family when losing. The conclusion is, that in determining the rationality of a gambler's decision it is to be considered, that losing a certain amount of money is not necessarily as "good" as losing the same amount of money is "bad".

In the case of this very game: when you ask the question: is it rational for me to pay x amount of money to play the game, the decision can only be made considering the exact chances of losing or winning any of the possible amounts of money. And this desicion cannot be made, if my wealth is not considered, and if my general well-being in function of my wealth is not considered in its very complexity.

This "paradox" can help to urge ourselves, to take the question of rational decision-making in its real depth and complexity.

The "paradox" by the way lies in the question of this simplified rationality-definition. The odds of winning in itself does not present any paradox: if played infinite times, any fee be it ever so large will indeed pay off eventually. [Edit:] The point is, that you only can play it _once_, and that's why you cannot decide without considering your wealth, for example: you cannot decide as if you were an investment banker with unlimited funds behind you, who doesn't have to worry about the end-of-month balance, because the balance is taken at the end of all times, infinitely in the future.